Simulation Study

Simulation 1

Case 1: \(n > p\) - adaptive greedy

simu1_part1_ada <- run_simulation(nb_simu = 500, true_beta = c(1, rep(0,58), 1),
                              n = 80, n_star = 800, sigma = 1, cor_X = 0.5,
                              xlim = c(1.035, 1.095), ylim = c(0.02, 0.1),
                              xlab = expression(PE[y]), 
                              ylab = expression(MSE[beta]),
                              title = "Simulation 1: n > p",
                              adaptive = TRUE)

kable(simu1_part1_ada)  

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	3.924397	0.0520607	4.7643774	0.0788026	0.0	100.0	2.000	58.000	60.000
AIC	1.581485	0.0258145	0.7567005	0.0502997	5.4	99.0	1.990	10.520	12.510
BIC	1.080430	0.0060340	0.0782325	0.0091298	49.4	98.8	1.988	1.164	3.152
HQ	1.216138	0.0196712	0.2371755	0.0217632	21.2	98.8	1.988	3.976	5.964
lasso	1.129553	0.0054523	0.1352864	0.0038669	3.4	100.0	2.000	8.324	10.324
PDC	1.057575	0.0034222	0.0429433	0.0053703	59.8	98.0	1.978	0.520	2.498
enet	1.166245	0.0074518	0.1749347	0.0061593	2.0	100.0	2.000	11.168	13.168
alasso	1.045711	0.0037980	0.0354535	0.0025080	67.8	100.0	2.000	0.936	2.936
MCP	1.072988	0.0051792	0.0298685	0.0021546	56.4	100.0	2.000	1.350	3.350
SCAD	1.078244	0.0041694	0.0327290	0.0022017	25.0	100.0	2.000	2.904	4.904

Case 2: \(n > p\) - non-adaptive greedy

simu1_part1_non_ada <- run_simulation(nb_simu = 500, true_beta = c(1, rep(0,58), 1),
                              n = 80, n_star = 800, sigma = 1, cor_X = 0.5,
                              xlim = c(1.035, 1.095), ylim = c(0.02, 0.1),
                              xlab = expression(PE[y]),
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 1: n > p",
                              adaptive = FALSE) 

kable(simu1_part1_non_ada) 

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	3.924397	0.0520607	4.7643774	0.0788026	0.0	100	2	58.000	60.000
AIC	1.514825	0.0232021	0.6381180	0.0309768	1.0	100	2	9.250	11.250
BIC	1.134133	0.0093528	0.1333654	0.0083030	27.0	100	2	1.742	3.742
HQ	1.292105	0.0114149	0.3335967	0.0174288	7.8	100	2	4.584	6.584
lasso	1.129553	0.0054523	0.1352864	0.0038669	3.4	100	2	8.324	10.324
PDC	1.066817	0.0045007	0.0557219	0.0073714	56.2	100	2	0.588	2.588
enet	1.166245	0.0074518	0.1749347	0.0061593	2.0	100	2	11.168	13.168
alasso	1.045711	0.0037980	0.0354535	0.0025080	67.8	100	2	0.936	2.936
MCP	1.072988	0.0051792	0.0298685	0.0021546	56.4	100	2	1.350	3.350
SCAD	1.078244	0.0041694	0.0327290	0.0022017	25.0	100	2	2.904	4.904

Case 3: \(n < p\) - adaptive greedy

simu1_part2_ada <- run_simulation(nb_simu = 500, true_beta = c(1, rep(0, 58), 1, rep(0, 100)),
                              n = 80, n_star = 800, sigma = 1, cor_X = 0.5,
                              xlim = c(1.045,1.22), ylim = c(0.03,0.23),
                              xlab = expression(PE[y]), 
                              ylab = expression(MSE[beta]),
                              title = "Simulation 1: n < p",
                              adaptive = TRUE)

kable(simu1_part2_ada) 

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	1.785228	0.0123505	0.9016844	0.0170075	0.0	100	2	31.000	33.000
AIC	1.452693	0.0131253	0.4715656	0.0142591	1.0	100	2	6.342	8.342
BIC	1.198411	0.0085915	0.1957703	0.0106081	22.2	100	2	2.104	4.104
HQ	1.323303	0.0130418	0.3553142	0.0136168	4.4	100	2	4.172	6.172
lasso	1.175958	0.0078497	0.1829157	0.0052767	1.6	100	2	11.778	13.778
PDC	1.090815	0.0062457	0.0916089	0.0065994	47.6	100	2	0.798	2.798
enet	1.218933	0.0072229	0.2385542	0.0097881	0.6	100	2	16.204	18.204
alasso	1.056450	0.0043868	0.0409949	0.0030151	63.6	100	2	1.106	3.106
MCP	1.081824	0.0034671	0.0394910	0.0035240	46.6	100	2	1.970	3.970
SCAD	1.090216	0.0045551	0.0492014	0.0029099	12.4	100	2	5.564	7.564

Case 4: \(n < p\) - non-adaptive greedy

simu1_part2_non_ada <- run_simulation(nb_simu = 500, true_beta = c(1, rep(0, 58), 1, rep(0, 100)),
                              n = 80, n_star = 800, sigma = 1, cor_X = 0.5,
                              xlim = c(1.045,1.22), ylim = c(0.03,0.23),
                              xlab = expression(PE[y]), 
                              ylab = expression(MSE[beta]),
                              title = "Simulation 1: n < p",
                              adaptive = FALSE)

kable(simu1_part2_non_ada) 

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	1.785228	0.0123505	0.9016844	0.0170075	0.0	100	2	31.000	33.000
AIC	1.385566	0.0122123	0.4218037	0.0112756	0.0	100	2	6.050	8.050
BIC	1.198981	0.0065862	0.1960937	0.0092454	14.8	100	2	2.128	4.128
HQ	1.299758	0.0092714	0.3113019	0.0125721	2.8	100	2	4.030	6.030
lasso	1.175958	0.0078497	0.1829157	0.0052767	1.6	100	2	11.778	13.778
PDC	1.098539	0.0062526	0.1015345	0.0057471	42.0	100	2	0.890	2.890
enet	1.218933	0.0072229	0.2385542	0.0097881	0.6	100	2	16.204	18.204
alasso	1.056450	0.0043868	0.0409949	0.0030151	63.6	100	2	1.106	3.106
MCP	1.081824	0.0034671	0.0394910	0.0035240	46.6	100	2	1.970	3.970
SCAD	1.090216	0.0045551	0.0492014	0.0029099	12.4	100	2	5.564	7.564

Simulation 2

Case 1: \(n > p\) - adaptive greedy

simu2_part1_ada <- run_simulation(nb_simu = 500, true_beta = rep(c(0.3, 0), 5),
                              n = 100, n_star = 1000, sigma = 1, cor_X = 0.75,
                              xlim = c(1.065,1.17), ylim = c(0.18,0.46),
                              xlab = expression(PE[y]),
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 2: n > p",
                              adaptive = TRUE)

kable(simu2_part1_ada) 

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	1.107841	0.0035288	0.3329839	0.0121383	0.0	100.0	5.000	5.000	10.000
AIC	1.124838	0.0038600	0.3514957	0.0147284	3.8	7.8	3.134	0.856	3.990
BIC	1.155585	0.0044200	0.4046674	0.0210016	0.4	0.6	2.586	0.532	3.118
HQ	1.136904	0.0042594	0.3599505	0.0132298	1.4	1.8	2.868	0.658	3.526
lasso	1.078078	0.0037695	0.2066077	0.0078130	2.6	63.0	4.556	2.680	7.236
PDC	1.144741	0.0047182	0.3735564	0.0156125	0.4	0.6	2.692	0.568	3.260
enet	1.075393	0.0030004	0.1956800	0.0080185	0.8	71.8	4.680	3.054	7.734
alasso	1.091444	0.0040757	0.2487335	0.0112976	5.8	37.0	4.038	1.794	5.832
MCP	1.138029	0.0032298	0.3635244	0.0172100	5.2	34.8	3.772	2.018	5.790
SCAD	1.136440	0.0039713	0.3589930	0.0131578	5.8	37.4	3.902	2.174	6.076

Case 2: \(n > p\) - non-adaptive greedy

simu2_part1_nonada <- run_simulation(nb_simu = 500, true_beta = rep(c(0.3, 0), 5),
                              n = 100, n_star = 1000, sigma = 1, cor_X = 0.75,
                              xlim = c(1.065,1.17), ylim = c(0.18,0.46),
                              xlab = expression(PE[y]),
                              ylab = expression(MSE[beta]),
                              title = "Simulation 2: n > p", 
                              adaptive = FALSE)

kable(simu2_part1_nonada) 

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	1.107841	0.0035288	0.3329839	0.0121383	0.0	100.0	5.000	5.000	10.000
AIC	1.125236	0.0036476	0.3624555	0.0115668	2.2	6.4	3.006	1.082	4.088
BIC	1.159894	0.0035174	0.4539168	0.0173506	0.4	0.6	2.374	0.758	3.132
HQ	1.135682	0.0043017	0.3898845	0.0158302	2.0	2.8	2.708	0.870	3.578
lasso	1.078078	0.0037695	0.2066077	0.0078130	2.6	63.0	4.556	2.680	7.236
PDC	1.144557	0.0041778	0.4204443	0.0186504	1.4	1.6	2.502	0.824	3.326
enet	1.075393	0.0030004	0.1956800	0.0080185	0.8	71.8	4.680	3.054	7.734
alasso	1.091444	0.0040757	0.2487335	0.0112976	5.8	37.0	4.038	1.794	5.832
MCP	1.138029	0.0032298	0.3635244	0.0172100	5.2	34.8	3.772	2.018	5.790
SCAD	1.136440	0.0039713	0.3589930	0.0131578	5.8	37.4	3.902	2.174	6.076

Case 3: \(n < p\) - adaptive

simu2_part2_ada <- run_simulation(nb_simu = 500, 
                                     true_beta = c(rep(c(0.3, 0), 5), rep(0, 100)),
                              n = 100, n_star = 1000, sigma = 1, cor_X = 0.75,
                              xlim = c(1.14,1.37), ylim = c(0.21,0.74),
                              xlab = expression(PE[y]),
                              ylab = expression(MSE[beta]),
                              title = "Simulation 2: n < p", 
                              adaptive = TRUE)

kable(simu2_part2_ada) 

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	1.650416	0.0101951	1.5687304	0.0341128	0.0	99.8	4.998	34.002	39.000
AIC	1.403845	0.0100389	0.7874925	0.0253773	0.0	3.8	2.838	6.016	8.854
BIC	1.291410	0.0060477	0.6117670	0.0113472	0.0	0.2	2.316	2.230	4.546
HQ	1.340386	0.0084212	0.6765585	0.0174947	0.0	1.6	2.626	4.040	6.666
lasso	1.165019	0.0055075	0.2577517	0.0078091	0.6	43.4	4.240	9.760	14.000
PDC	1.271483	0.0053393	0.5808807	0.0150697	0.0	0.0	1.974	1.098	3.072
enet	1.160756	0.0045681	0.2413781	0.0049370	0.2	60.2	4.514	13.892	18.406
alasso	1.183282	0.0060611	0.3603656	0.0137988	1.0	10.4	3.064	2.436	5.500
MCP	1.283560	0.0057373	0.5832792	0.0159249	0.2	0.8	2.174	3.558	5.732
SCAD	1.269043	0.0063063	0.4602072	0.0091509	0.8	10.0	3.040	5.790	8.830

Case 4: \(n < p\) - non-adaptive

simu2_part2_non_ada <- run_simulation(nb_simu = 500, 
                                     true_beta = c(rep(c(0.3, 0), 5), rep(0, 100)),
                              n = 100, n_star = 1000, sigma = 1, cor_X = 0.75,
                              xlim = c(1.14,1.37), ylim = c(0.21,0.74),
                              xlab = expression(PE[y]),
                              ylab = expression(MSE[beta]),
                              title = "Simulation 2: n < p", 
                              adaptive = FALSE)

kable(simu2_part2_non_ada) 

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	1.650416	0.0101951	1.5687304	0.0341128	0.0	99.8	4.998	34.002	39.000
AIC	1.356395	0.0088333	0.6652823	0.0183908	0.0	3.6	2.746	5.622	8.368
BIC	1.278110	0.0077329	0.6015461	0.0102443	0.2	0.4	2.180	2.184	4.364
HQ	1.319665	0.0089797	0.6172520	0.0133371	0.2	2.0	2.488	3.792	6.280
lasso	1.165019	0.0055075	0.2577517	0.0078091	0.6	43.4	4.240	9.760	14.000
PDC	1.265392	0.0077299	0.5991625	0.0114898	0.2	0.2	1.864	1.192	3.056
enet	1.160756	0.0045681	0.2413781	0.0049370	0.2	60.2	4.514	13.892	18.406
alasso	1.183282	0.0060611	0.3603656	0.0137988	1.0	10.4	3.064	2.436	5.500
MCP	1.283560	0.0057373	0.5832792	0.0159249	0.2	0.8	2.174	3.558	5.732
SCAD	1.269043	0.0063063	0.4602072	0.0091509	0.8	10.0	3.040	5.790	8.830

Simulation 3

Case 1: \(n > p\) - adaptive greedy

beta <- c(rep(c(2,0,1),2),rep(0,16),rep(0.1,6),rep(0,16),rep(c(2,0,1),2))
simu3_part1_ada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 2, cor_X = 0.5,
                              xlim = c(4.73,5.4), ylim = c(0.79,1.8),
                              xlab = expression(PE[y]),
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 3: n > p",
                              adaptive = TRUE)

kable(simu3_part1_ada) 

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	7.828768	0.0539561	6.3269382	0.1089886	0	100.0	14.000	36.000	50.000
AIC	5.602199	0.0407343	2.3001445	0.1069784	0	0.0	8.946	4.972	13.918
BIC	4.984983	0.0364475	1.1753320	0.0678564	0	0.0	8.086	1.230	9.316
HQ	5.217643	0.0629048	1.5831001	0.0692687	0	0.0	8.502	2.742	11.244
lasso	5.067051	0.0277870	1.2319757	0.0320709	0	0.2	10.146	9.974	20.120
PDC	5.048466	0.0509589	1.3548679	0.0735508	0	0.0	7.392	0.360	7.752
enet	5.126117	0.0249846	1.3744623	0.0363623	0	1.8	10.474	12.024	22.498
alasso	4.803297	0.0314939	0.9325193	0.0448793	0	0.0	8.678	2.524	11.202
MCP	5.233635	0.0527333	1.0135877	0.0516727	0	0.0	8.730	2.936	11.666
SCAD	5.201045	0.0463538	1.0172687	0.0584409	0	0.0	9.450	5.480	14.930

Case 2: \(n > p\) - non-adaptive greedy

beta <- c(rep(c(2,0,1),2),rep(0,16),rep(0.1,6),rep(0,16),rep(c(2,0,1),2))
simu3_part1_nonada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 2, cor_X = 0.5,
                              xlim = c(4.73,5.4), ylim = c(0.79,1.8),
                              xlab = expression(PE[y]),
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 3: n > p",
                              adaptive = FALSE)

kable(simu3_part1_nonada) 

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	7.828768	0.0539561	6.3269382	0.1089886	0	100.0	14.000	36.000	50.000
AIC	5.471716	0.0418450	1.8685351	0.0647247	0	0.0	9.028	4.572	13.600
BIC	4.913370	0.0289821	1.0343625	0.0446144	0	0.0	8.198	1.196	9.394
HQ	5.133586	0.0497034	1.4275142	0.0659176	0	0.0	8.612	2.578	11.190
lasso	5.067051	0.0277870	1.2319757	0.0320709	0	0.2	10.146	9.974	20.120
PDC	4.996859	0.0439417	1.2350775	0.0902718	0	0.0	7.558	0.526	8.084
enet	5.126117	0.0249846	1.3744623	0.0363623	0	1.8	10.474	12.024	22.498
alasso	4.803297	0.0314939	0.9325193	0.0448793	0	0.0	8.678	2.524	11.202
MCP	5.233635	0.0527333	1.0135877	0.0516727	0	0.0	8.730	2.936	11.666
SCAD	5.201045	0.0463538	1.0172687	0.0584409	0	0.0	9.450	5.480	14.930

Case 3: \(n < p\) - adaptive

beta <- c(rep(c(2,0,1),2),rep(0,16),rep(0.1,6),rep(0,16),rep(c(2,0,1),2),
          rep(0,100))
simu3_part2_ada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 2, cor_X = 0.5,
                              xlim = c(4.85,5.9), ylim = c(1,2),
                              xlab = expression(PE[y]),
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 3: n < p",
                              adaptive = TRUE)

kable(simu3_part2_ada) 

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	6.981891	0.0610234	3.765791	0.0692500	0	0.4	9.138	29.862	39.000
AIC	5.638632	0.0468071	1.915825	0.0537250	0	0.0	8.098	4.624	12.722
BIC	5.058629	0.0493449	1.212897	0.0534257	0	0.0	7.786	1.370	9.156
HQ	5.326638	0.0583424	1.570145	0.0536998	0	0.0	7.954	2.734	10.688
lasso	5.635840	0.0346758	1.665008	0.0314404	0	0.0	9.334	18.934	28.268
PDC	5.061261	0.0674485	1.361253	0.1021703	0	0.0	7.264	0.564	7.828
enet	5.792712	0.0358061	1.841099	0.0443238	0	0.2	9.516	23.152	32.668
alasso	4.994438	0.0309109	1.134744	0.0420539	0	0.0	8.194	3.126	11.320
MCP	5.637378	0.0572940	1.459303	0.0736045	0	0.0	8.114	4.784	12.898
SCAD	5.592650	0.0564631	1.518394	0.0695838	0	0.0	8.888	11.420	20.308

Case 4: \(n < p\) - non-adaptive

beta <- c(rep(c(2,0,1),2),rep(0,16),rep(0.1,6),rep(0,16),rep(c(2,0,1),2),
          rep(0,100))
simu3_part2_non_ada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 2, cor_X = 0.5,
                              xlim = c(4.85,5.9), ylim = c(1,2),
                              xlab = expression(PE[y]),
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 3: n < p",
                              adaptive = FALSE)

kable(simu3_part2_non_ada) 

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	6.981891	0.0610234	3.765791	0.0692500	0	0.4	9.138	29.862	39.000
AIC	5.562745	0.0408023	1.821001	0.0558496	0	0.0	8.140	4.546	12.686
BIC	5.107543	0.0450648	1.264909	0.0722063	0	0.0	7.760	1.532	9.292
HQ	5.348961	0.0416001	1.540760	0.0444740	0	0.0	7.980	2.838	10.818
lasso	5.635840	0.0346758	1.665008	0.0314404	0	0.0	9.334	18.934	28.268
PDC	5.136446	0.0495950	1.457847	0.1109942	0	0.0	7.258	0.668	7.926
enet	5.792712	0.0358061	1.841099	0.0443238	0	0.2	9.516	23.152	32.668
alasso	4.994438	0.0309109	1.134744	0.0420539	0	0.0	8.194	3.126	11.320
MCP	5.637378	0.0572940	1.459303	0.0736045	0	0.0	8.114	4.784	12.898
SCAD	5.592650	0.0564631	1.518394	0.0695838	0	0.0	8.888	11.420	20.308

Simulation 4

Case 1: \(n > p\) - adaptive greedy

beta <- c(rep(c(2,0,1),2),rep(0,16),rep(0.1,6),rep(0,16),rep(c(2,0,1),2))
simu4_part1_ada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 2, cor_X = 0.5,
                              xlim = c(0.73,100.4), ylim = c(0.0079,20.8),   
                              xlab = expression(PE[y]), 
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 4: n > p",
                              adaptive = TRUE, omitted_variables = 1)

kable(simu4_part1_ada) 

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	13.502791	0.1501429	10.707603	0.1837823	0	0	49.000	0	49.000
AIC	10.171847	0.0852397	4.680953	0.1966095	0	0	13.262	0	13.262
BIC	9.288934	0.0568300	3.237664	0.1084220	0	0	8.066	0	8.066
HQ	9.509905	0.0842522	3.468800	0.1150142	0	0	10.158	0	10.158
lasso	8.678201	0.0460419	2.015333	0.0621440	0	0	19.092	0	19.092
PDC	9.696391	0.0804393	3.886826	0.1249451	0	0	6.090	0	6.090
enet	8.814902	0.0393679	2.197442	0.0566599	0	0	21.730	0	21.730
alasso	8.614828	0.0564466	2.151635	0.0899776	0	0	10.972	0	10.972
MCP	9.477908	0.0819999	3.036092	0.0992974	0	0	11.008	0	11.008
SCAD	9.378062	0.0614218	2.977569	0.0896044	0	0	14.468	0	14.468

Case 2: \(n > p\) - nonadaptive greedy

beta <- c(rep(c(2,0,1),2),rep(0,16),rep(0.1,6),rep(0,16),rep(c(2,0,1),2))
simu4_part1_nonada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 2, cor_X = 0.5,
                              xlim = c(0.73,100.4), ylim = c(0.0079,20.8),   
                              xlab = expression(PE[y]), 
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 4: n > p",
                              adaptive = FALSE, omitted_variables = 1)

kable(simu4_part1_nonada) 

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	13.502791	0.1501429	10.707603	0.1837823	0	0	49.000	0	49.000
AIC	9.793129	0.0768132	3.784237	0.1184813	0	0	12.892	0	12.892
BIC	9.241664	0.0697646	3.055688	0.0801707	0	0	8.176	0	8.176
HQ	9.456914	0.0992046	3.329390	0.0972033	0	0	10.380	0	10.380
lasso	8.678201	0.0460419	2.015333	0.0621440	0	0	19.092	0	19.092
PDC	9.651446	0.0670652	3.882546	0.0998996	0	0	6.306	0	6.306
enet	8.814902	0.0393679	2.197442	0.0566599	0	0	21.730	0	21.730
alasso	8.614828	0.0564466	2.151635	0.0899776	0	0	10.972	0	10.972
MCP	9.477908	0.0819999	3.036092	0.0992974	0	0	11.008	0	11.008
SCAD	9.378062	0.0614218	2.977569	0.0896044	0	0	14.468	0	14.468

Case 3: \(n < p\) - adaptive greedy

beta <- c(rep(c(2,0,1),2),rep(0,16),rep(0.1,6),rep(0,16),rep(c(2,0,1),2), rep(0, 100))
simu4_part2_ada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 2, cor_X = 0.5,
                              xlim = c(0.73,100.4), ylim = c(0.0079,20.8),   
                              xlab = expression(PE[y]), 
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 4: n < p",
                              adaptive = TRUE, omitted_variables = 1)

kable(simu4_part2_ada) 

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	11.993168	0.0918683	6.166943	0.0984212	0	0	39.000	0	39.000
AIC	10.267386	0.0995894	3.877167	0.1223970	0	0	12.792	0	12.792
BIC	9.653047	0.0716499	3.379400	0.0926498	0	0	8.548	0	8.548
HQ	9.807477	0.0833390	3.467016	0.0991983	0	0	10.586	0	10.586
lasso	9.581549	0.0545424	2.682683	0.0620017	0	0	26.404	0	26.404
PDC	9.871702	0.0776319	3.934555	0.0797900	0	0	6.214	0	6.214
enet	9.754135	0.0581014	2.806577	0.0570407	0	0	31.418	0	31.418
alasso	9.075474	0.0560001	2.579230	0.0734337	0	0	10.940	0	10.940
MCP	10.184737	0.0721937	4.065031	0.1145714	0	0	11.216	0	11.216
SCAD	10.142393	0.0667124	3.857304	0.0715728	0	0	18.320	0	18.320

Case 4: \(n < p\) - nonadaptive greedy

beta <- c(rep(c(2,0,1),2),rep(0,16),rep(0.1,6),rep(0,16),rep(c(2,0,1),2), rep(0, 100))
simu4_part2_nonada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 2, cor_X = 0.5,
                              xlim = c(0.73,100.4), ylim = c(0.0079,20.8),   
                              xlab = expression(PE[y]), 
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 4: n < p",
                              adaptive = FALSE, omitted_variables = 1)

kable(simu4_part2_nonada) 

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	11.993168	0.0918683	6.166943	0.0984212	0	0	39.000	0	39.000
AIC	10.147548	0.0615976	3.783625	0.0653097	0	0	12.626	0	12.626
BIC	9.672819	0.0627906	3.512746	0.1176478	0	0	8.550	0	8.550
HQ	9.829220	0.0576269	3.497726	0.1063502	0	0	10.550	0	10.550
lasso	9.581549	0.0545424	2.682683	0.0620017	0	0	26.404	0	26.404
PDC	9.951037	0.0692510	4.057780	0.1249516	0	0	6.380	0	6.380
enet	9.754135	0.0581014	2.806577	0.0570407	0	0	31.418	0	31.418
alasso	9.075474	0.0560001	2.579230	0.0734337	0	0	10.940	0	10.940
MCP	10.184737	0.0721937	4.065031	0.1145714	0	0	11.216	0	11.216
SCAD	10.142393	0.0667124	3.857304	0.0715728	0	0	18.320	0	18.320

Simulation 5

Case 1: \(n > p\) - adaptive greedy

beta <- c(3,rep(0,5),1,rep(0,10),rep(0.05,3),rep(0,10),2,rep(0,20))
simu5_part1_ada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 3, cor_X = 0.25,
                              xlim = c(19.3,20.2), ylim = c(1.6,2.75), 
                              xlab = expression(PE[y]), 
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 5: n > p",
                              adaptive = TRUE, omitted_variables = 1)

kable(simu5_part1_ada)  

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	35.21969	0.3187061	19.611517	0.3362240	0	0	50.000	0	50.000
AIC	23.81532	0.2259463	6.564957	0.2060844	0	0	8.434	0	8.434
BIC	19.94307	0.0804733	2.505281	0.1080523	0	0	2.748	0	2.748
HQ	21.53516	0.1586480	4.101247	0.1566812	0	0	4.952	0	4.952
lasso	19.67917	0.0716364	2.307344	0.0708563	0	0	8.512	0	8.512
PDC	19.57266	0.0811335	1.807336	0.0602241	0	0	1.878	0	1.878
enet	19.85348	0.0655296	2.461787	0.0721343	0	0	11.622	0	11.622
alasso	19.63743	0.0829160	2.005477	0.0491488	0	0	2.940	0	2.940
MCP	19.82289	0.0748150	1.979799	0.0488847	0	0	4.310	0	4.310
SCAD	19.80302	0.1020472	2.085949	0.0606429	0	0	6.824	0	6.824

Case 2: \(n > p\) - nonadaptive greedy

beta <- c(3,rep(0,5),1,rep(0,10),rep(0.05,3),rep(0,10),2,rep(0,20))
simu5_part1_nonada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 3, cor_X = 0.25,
                              xlim = c(19.3,20.2), ylim = c(1.6,2.75), 
                              xlab = expression(PE[y]), 
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 5: n > p",
                              adaptive = FALSE, omitted_variables = 1)

kable(simu5_part1_nonada)  

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	35.21969	0.3187061	19.611517	0.3362240	0	0	50.000	0	50.000
AIC	23.20874	0.2026071	5.898750	0.2338397	0	0	8.110	0	8.110
BIC	19.95860	0.0791702	2.551035	0.0919496	0	0	2.908	0	2.908
HQ	21.28517	0.1197085	4.073351	0.1319092	0	0	4.990	0	4.990
lasso	19.67917	0.0716364	2.307344	0.0708563	0	0	8.512	0	8.512
PDC	19.56326	0.0909917	1.778441	0.0453945	0	0	1.916	0	1.916
enet	19.85348	0.0655296	2.461787	0.0721343	0	0	11.622	0	11.622
alasso	19.63743	0.0829160	2.005477	0.0491488	0	0	2.940	0	2.940
MCP	19.82289	0.0748150	1.979799	0.0488847	0	0	4.310	0	4.310
SCAD	19.80302	0.1020472	2.085949	0.0606429	0	0	6.824	0	6.824

Case 3: \(n < p\) - adaptive greedy

beta <- c(3,rep(0,5),1,rep(0,10),rep(0.05,3),rep(0,10),2,rep(0,120))
simu5_part2_ada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 3, cor_X = 0.25,
                              xlim = c(19.3,22), ylim = c(2,3.5),
                              xlab = expression(PE[y]), 
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 5: n < p",
                              adaptive = TRUE, omitted_variables = 1)

kable(simu5_part2_ada)  

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	29.60585	0.2358299	12.311953	0.1829561	0	0	39.000	0	39.000
AIC	28.39794	0.1778518	11.129597	0.1605614	0	0	13.998	0	13.998
BIC	24.27648	0.2847263	6.810123	0.2441391	0	0	6.094	0	6.094
HQ	27.26201	0.1816258	10.122818	0.1832957	0	0	10.514	0	10.514
lasso	20.17443	0.1012597	2.725528	0.0690565	0	0	11.882	0	11.882
PDC	20.43240	0.0999867	2.992834	0.1206886	0	0	2.208	0	2.208
enet	20.41452	0.0936276	2.961706	0.0704891	0	0	17.166	0	17.166
alasso	19.87138	0.1050836	2.211785	0.0854961	0	0	2.844	0	2.844
MCP	19.99677	0.0954209	2.200610	0.0613539	0	0	4.478	0	4.478
SCAD	20.18132	0.0859469	2.539315	0.0528157	0	0	9.028	0	9.028

Case 4: \(n < p\) - nonadaptive greedy

beta <- c(3,rep(0,5),1,rep(0,10),rep(0.05,3),rep(0,10),2,rep(0,120))
simu5_part2_nonada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 3, cor_X = 0.25,
                              xlim = c(19.3,22), ylim = c(2,3.5),
                              xlab = expression(PE[y]), 
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 5: n < p",
                              adaptive = FALSE, omitted_variables = 1)

kable(simu5_part2_nonada)  

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	29.60585	0.2358299	12.311953	0.1829561	0	0	39.000	0	39.000
AIC	27.33470	0.1732158	10.041626	0.1523226	0	0	13.372	0	13.372
BIC	23.35517	0.2004715	5.756583	0.2621362	0	0	5.708	0	5.708
HQ	25.95172	0.1607828	8.642288	0.1559651	0	0	9.788	0	9.788
lasso	20.17443	0.1012597	2.725528	0.0690565	0	0	11.882	0	11.882
PDC	20.28612	0.0818379	2.731973	0.1146394	0	0	2.362	0	2.362
enet	20.41452	0.0936276	2.961706	0.0704891	0	0	17.166	0	17.166
alasso	19.87138	0.1050836	2.211785	0.0854961	0	0	2.844	0	2.844
MCP	19.99677	0.0954209	2.200610	0.0613539	0	0	4.478	0	4.478
SCAD	20.18132	0.0859469	2.539315	0.0528157	0	0	9.028	0	9.028

Simulation 6

Case 1: \(n > p\) - adaptive greedy

beta <- c(3,rep(0,5),1,rep(0,10),rep(0.05,3),rep(0,10),2,rep(0,20))
simu6_part1_ada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 3, cor_X = 0.25,
                              xlab = expression(PE[y]), 
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 6: n > p",
                              adaptive = TRUE)

kable(simu6_part1_ada)  

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	18.164562	0.1888375	10.4573591	0.1817820	0	100.0	6.000	45.000	51.000
AIC	12.128975	0.1194250	3.2084795	0.1279782	0	0.6	3.222	5.876	9.098
BIC	10.185307	0.0635619	1.1082839	0.0371106	0	0.0	2.804	1.084	3.888
HQ	10.865372	0.0800132	1.9031863	0.1085088	0	0.2	2.976	2.820	5.796
lasso	10.335688	0.0426948	1.3245968	0.0322219	0	1.6	3.446	7.608	11.054
PDC	9.967106	0.0555835	1.0096835	0.0413893	0	0.0	2.668	0.506	3.174
enet	10.550470	0.0426769	1.5172023	0.0412639	0	3.0	3.618	10.290	13.908
alasso	9.928824	0.0503249	0.9591637	0.0485525	0	0.2	2.798	1.298	4.096
MCP	10.071255	0.0448669	0.7723096	0.0387571	0	0.0	2.944	1.968	4.912
SCAD	10.073702	0.0494998	0.7556559	0.0294226	0	0.0	3.156	4.350	7.506

Case 2: \(n > p\) - nonadaptive greedy

beta <- c(3,rep(0,5),1,rep(0,10),rep(0.05,3),rep(0,10),2,rep(0,20))
simu6_part1_nonada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 3, cor_X = 0.25,
                              xlab = expression(PE[y]), 
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 6: n > p",
                              adaptive = FALSE)

kable(simu6_part1_nonada)  

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	18.164562	0.1888375	10.4573591	0.1817820	0	100.0	6.000	45.000	51.000
AIC	11.983331	0.1228147	3.0357047	0.0768831	0	0.2	3.276	5.748	9.024
BIC	10.186848	0.0754578	1.1317777	0.0417861	0	0.0	2.882	1.180	4.062
HQ	10.968588	0.0926223	1.9411459	0.0960494	0	0.0	3.062	2.834	5.896
lasso	10.335688	0.0426948	1.3245968	0.0322219	0	1.6	3.446	7.608	11.054
PDC	9.856713	0.0555075	0.7599161	0.0909081	0	0.0	2.722	0.394	3.116
enet	10.550470	0.0426769	1.5172023	0.0412639	0	3.0	3.618	10.290	13.908
alasso	9.928824	0.0503249	0.9591637	0.0485525	0	0.2	2.798	1.298	4.096
MCP	10.071255	0.0448669	0.7723096	0.0387571	0	0.0	2.944	1.968	4.912
SCAD	10.073702	0.0494998	0.7556559	0.0294226	0	0.0	3.156	4.350	7.506

Case 3: \(n < p\) - adaptive greedy

beta <- c(3,rep(0,5),1,rep(0,10),rep(0.05,3),rep(0,10),2,rep(0,120))
simu6_part2_ada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 3, cor_X = 0.25,
                              xlab = expression(PE[y]), 
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 6: n < p",
                              adaptive = TRUE)

kable(simu6_part2_ada)  

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	15.75937	0.0977847	6.924629	0.1181671	0	0.2	3.518	35.482	39.000
AIC	13.41529	0.1168050	4.560226	0.1222878	0	0.0	2.920	8.308	11.228
BIC	11.08738	0.1000759	2.087393	0.0904802	0	0.0	2.714	2.534	5.248
HQ	12.42498	0.1029060	3.394891	0.1293898	0	0.0	2.838	5.378	8.216
lasso	10.84788	0.0486911	1.797697	0.0535625	0	0.0	3.124	12.774	15.898
PDC	10.23524	0.0701200	1.187085	0.0278475	0	0.0	2.552	0.776	3.328
enet	11.05887	0.0551945	2.096985	0.0708282	0	0.0	3.222	16.416	19.638
alasso	10.05894	0.0606301	1.125238	0.0255999	0	0.0	2.576	1.500	4.076
MCP	10.20083	0.0464832	1.027896	0.0294854	0	0.0	2.754	3.040	5.794
SCAD	10.25321	0.0529682	1.030886	0.0228934	0	0.0	2.984	7.648	10.632

Case 4: \(n < p\) - nonadaptive greedy

beta <- c(3,rep(0,5),1,rep(0,10),rep(0.05,3),rep(0,10),2,rep(0,120))
simu6_part2_nonada <- run_simulation(nb_simu = 500, true_beta = beta,
                              n = 100, n_star = 1000, sigma = 3, cor_X = 0.25,
                              xlab = expression(PE[y]), 
                              ylab = expression(MSE[beta]), 
                              title = "Simulation 6: n < p",
                              adaptive = FALSE)

kable(simu6_part2_nonada)  

	Median MSEy	SD Med MSEy	Median MSEb	SD Med MSEb	correct	included	true+	false+	# sel.
LS	15.75937	0.0977847	6.924629	0.1181671	0	0.2	3.518	35.482	39.000
AIC	12.99853	0.1354686	3.972303	0.1060509	0	0.0	2.930	7.844	10.774
BIC	11.04486	0.0795333	2.013169	0.0826483	0	0.0	2.736	2.500	5.236
HQ	12.14950	0.1096684	3.169140	0.1036183	0	0.0	2.842	5.024	7.866
lasso	10.84788	0.0486911	1.797697	0.0535625	0	0.0	3.124	12.774	15.898
PDC	10.26517	0.0819376	1.225533	0.0295604	0	0.0	2.602	0.882	3.484
enet	11.05887	0.0551945	2.096985	0.0708282	0	0.0	3.222	16.416	19.638
alasso	10.05894	0.0606301	1.125238	0.0255999	0	0.0	2.576	1.500	4.076
MCP	10.20083	0.0464832	1.027896	0.0294854	0	0.0	2.754	3.040	5.794
SCAD	10.25321	0.0529682	1.030886	0.0228934	0	0.0	2.984	7.648	10.632